OpenAI has announced a solution to the Navier–Stokes existence and smoothness problem, one of the Millennium Prize Problems. This proof demonstrates that the dynamics of the Navier-Stokes equations for fluid motion can develop a singularity in finite time.

The proof was generated by an internal OpenAI system, which is significantly more capable than GPT-6 Astra, and both a writeup and a formalization in Lean have been shared. This resolves a profound mathematical question, unresolved for approximately 90 years, regarding whether the continuum approximation of fluid can break down.

The solution describes a vortex that spirals inward and elongates, like spaghetti, with its central region shrinking while accelerating, yet maintaining finite energy as it forms a singularity. This resolves statements “C” and “D” in the official Millennium Prize formulation of the Navier–Stokes problem.

OpenAI has been training a new internal model since August 28, 2026, which has shown unprecedented performance in benchmarks, including mathematics, with its training still ongoing. A system of coordinating AI agents, powered by this internal model and equipped with tools like internet access and code execution, was used to derive the proof.

A group involving on the order of 10,000 concurrent agents contributed to the Navier–Stokes resolution. The agents arrived at their resolution on September 5, 2026, with subsequent Lean formalization and verification taking an additional 17 hours via GPT-6 Astra.

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Meta has unveiled a new personal AI agent named Muse, which places a strong emphasis on user privacy protection. This agent aims to provide personalized assistance while securely managing individual data.

Muse is expected to assist users with daily tasks, offering a wide range of functionalities including information retrieval, content generation, and schedule management. Notably, privacy has been considered from the design phase, with an emphasis on transparency regarding data collection and usage.

This approach serves as a crucial differentiator for gaining user trust as AI agents become more widespread. Through Muse, Meta seeks to deliver Generative AI technology in a safer and more responsible manner.

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At Apple's September 2026 event, the highly anticipated folding iPhone was unveiled, drawing significant attention across the technology industry. This new device is expected to offer an innovative form factor and an enhanced user experience.

The event also showcased the Apple Watch Series 12, hinting at further advancements in health monitoring features and improved battery life. Additionally, the latest models of the popular wireless earbuds, AirPods, were announced, featuring enhancements in sound quality and connectivity.

These new products reaffirm Apple's commitment to enriching users' daily lives through the integration of hardware and software. The folding iPhone, in particular, holds the potential to redefine the future of smartphones.

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